Aspects of massive gauge theories on three sphere in infinite mass limit

We study the $S^3$ partition function of three-dimensional supersymmetric $\mathcal{N}=4$ U($N$) SQCD with massive matter multiplets in the infinite mass limit with the so-called Coulomb branch localization. We show that in the infinite mass limit the specific point of the Coulomb branch is chosen and contributes to the partition function dominantly. Therefore we can argue whether each multiplet included in the theory is effectively massless or not in this limit even on $S^3$ and conclude that the partition function becomes that of the effective theory on the specific point of the Coulomb branch in the infinite mass limit. In order to investigate which point of the Coulomb branch is dominant, we use the saddle point approximation in the large $N$ limit because the solution of the saddle point equation can be regarded as a specific point of the Coulomb branch. Then we calculate the partition functions for small rank $N$ and actually confirm that their behaviors in the infinite mass limit are consistent with the conjecture from the results in the large $N$ limit. Our result suggests that the partition function in the mass infinite limit corresponds to that of an interacting superconformal field theory.